THE DYNAMICS AND LIGHT CURVES OF BEAMED GAMMA RAY BURST AFTERGLOWS

Transcription

1 THE DYNAMICS AND LIGHT CURVES OF BEAMED GAMMA RAY BURST AFTERGLOWS James E. Rhoads Kitt Peak National Observatory, 950 North Cherry Avenue, Tucson, AZ Electronic mail: jrhoads@noao.edu Submitted to The Astrophysical Journal, 2/1998; current revision 3/1999 ABSTRACT The energy requirements of gamma ray bursts have in past been poorly constrained because of three major uncertainties: The distances to bursts, the degree of burst beaming, and the efficiency of gamma ray production. The first of these has been resolved, with both indirect evidence the distribution of bursts in flux and position) and direct evidence redshifted absorption features in the afterglow spectrum of GRB ) pointing to cosmological distances. We now wish to address the second uncertainty. Afterglows allow a statistical test of beaming, described in an earlier paper. In this paper, we modify a standard fireball afterglow model to explore the effects of beaming on burst remnant dynamics and afterglow emission. If the burst ejecta are beamed into angle ζ m, the burst remnant s evolution changes qualitatively once its bulk Lorentz factor Γ < 1/ζ m : Before this, Γ declines as a power law of radius, while afterwards, it declines exponentially. This change results in a broken power law light curve whose late-time decay is faster than expected for a purely spherical geometry. These predictions disagree with afterglow observations of GRB We explored several variations on our model, but none seems able to change this result. We therefore suggest that this burst is unlikely to have been highly beamed, and that its energy requirements were near those of isotropic models. More recent afterglows may offer the first practical applications for our beamed models. 1. INTRODUCTION Understanding the energy requirements and event rates of gamma ray bursts is necessary for any quantitative evaluation of a candidate burst progenitor. We need to know both how many progenitors we expect, and how much energy they need to produce in a single event. Until recently, both quantities were uncertain to 10 orders of magnitude because of the unknown distance to the bursts. The afterglow of GRB effectively ended that debate, because it showed absorption lines at a cosmological redshift z =0.835; Metzger et al 1997). This builds on earlier results from the Burst and Transient Source Experiment BATSE), which showed that the burst distribution on the sky is exquisitely isotropic while the distribution in flux is inhomogeneous Meegan et al 1996). These observations are best explained if the bursts are at cosmological distances. A very extended Galactic halo distribution might also work, but it would have to be unlike any other known population of Galactic objects. The isotropy is perhaps most important now for showing that multiple-population scenarios for gamma ray bursts cannot put any substantial fraction of the bursters at Galactic distances. It thus connects the GRB redshift bound to the vast majority of the burst population. The dominant remaining uncertainty in the bursters energy requirements is now whether the bursts radiate isotropically or are beamed into a very small solid angle. Such beaming is allowed though not required) by the gamma ray observations, because the ejecta from gamma ray bursts must be highly relativistic to explain the spectral properties of the emergent radiation Paczyński 1986, Goodman 1986), with inferred minimum Lorentz factors Γ > 100 Woods & Loeb 1995). The gamma rays we observe are therefore only those from material moving within angle 1/Γ of the line of sight, and offer no straightforward way of determining whether there are eject outside this narrow cone. These large Lorentz factors lead naturally to predictions of afterglow emission at longer wavelengths as the burst ejecta decelerate and interact with the surrounding material Paczyński & Rhoads 1993; Katz 1994; Mészáros & Rees 1997a). The characteristic frequency for this afterglow emission depends on the Lorentz factor of the burst remnant, and both decrease as the remnant evolves. Such models scored a recent triumph with the detection of X- ray, optical, and radio afterglows from gamma ray bursts GRBs) early in 1997 e.g., Costa et al 1997; van Paradijs et al 1997; Bond 1997; Frail et al 1997). The observed properties of the transients are in good overall agreement with the predictions of afterglow models Wijers, Rees, & Mészáros 1997; Waxman 1997a,b), although some worries remain Dar 1997). Because beaming depends on the relativistic nature of the flow, afterglows can be used to test the burst beaming hypothesis. At least two such tests are possible. First, because Γ is lower at the time of afterglow emission than during the GRB itself, the afterglow cannot be as collimated as the GRB can. This implies that the afterglow event rate should exceed the GRB event rate substantially if bursts are strongly beamed. Allowing for finite detection 1 Postal address: P.O. Box 26732, Tucson, AZ

2 2 Submitted to The Astrophysical Journal, 1998; current revision 3/1999 thresholds, N 12 Ω 1 N 1, 1) N 2 Ω 2 N 12 where N 1, N 2 are the measured event rates above our detection thresholds at our two frequencies; N 12 is the rate of events above threshold at both frequencies; and Ω 1,Ω 2 are the solid angles into which emission is beamed at the two frequencies. A full derivation of this result and discussion of its application is given in Rhoads 1997a). The second test is based on differences between the dynamical evolution of beamed and isotropic bursts. Burst ejecta decelerate through their interaction with the ambient medium. If the ejecta are initially beamed into a cone of opening angle ζ m, the deceleration changes qualitatively when the bulk Lorentz factor Γ drops to 1/ζ m. Prior to this, the working surface i.e. the area over which the expanding blast wave interacts with the surrounding medium) scales as r 2. At later times, the ejecta cloud has undergone significant lateral expansion in its frame, and the working surface increases more rapidly with r, eventually approaching an exponential growth. Spherical symmetry prevents this transition from occurring in unbeamed bursts. A brief analysis of this effect was presented in Rhoads 1997b). We have two major aims in this paper. First, we will present a full derivation of the late time burst remnant dynamics for a beamed gamma ray burst. We support this by calculating the emergent synchrotron radiation for two electron energy distribution models, but we do not attempt to do so for all possible fireball emission scenarios. Second, we observe that our model is not consistent with any small-angle beaming of GRB This implies a substantial minimum energy for this burst. If radiative efficiencies are lower than 10%, this limit approaches the maximum energy available in compact object merger events. We explore possible ways to evade this minimum energy requirement through other forms of beaming models, but find none. We therefore conjecture that such models cannot be constructed for GRB unless the usual fireball model assumptions about relativistic blast wave physics are substantially modified, and challenge the community to prove this assertion right or wrong. We explore the dynamical evolution of a model beamed burst in section 2. In section 3 we incorporate a model for the electron energy spectrum and magnetic field strength and so predict the emergent synchrotron radiation. In section 4, we compare the model with observed afterglows. The early 1997) data appeared inconsistent with the beaming model, suggesting that bursts are fairly isotropic and therefore very energetic events. Finally, in section 4.1, we explore variations on our model to try to reduce the inferred energy needs of GRB We comment briefly on more recent data and summarize our conclusions in section DYNAMICAL CONSEQUENCES OF BEAMING We explore the effects of beaming on burst evolution using the notation of Paczyński & Rhoads 1993). Let Γ 0 and M 0 be the initial Lorentz factor and ejecta mass, and ζ m the opening angle into which the ejecta move. The burst energy is E 0 =Γ 0 M 0 c 2. Let r be the radial coordinate in the burster frame; t, t co,andt thetimefrom the event measured in the burster frame, comoving ejecta frame, and terrestrial observer s frame; and f the ratio of swept up mass to M 0. The key assumptions in our beamed burst model are that 1) the initial energy and mass per unit solid angle are constant at angles θ<ζ m from the jet axis and zero for θ>ζ m ; 2) the total energy in the ejecta + sweptup material is approximately conserved; 3) the ambient medium has uniform density ρ; and 4) the cloud of ejecta + swept-up material expands in its comoving frame at the sound speed c s = c/ 3 appropriate for relativistic matter. The last of these assumptions implies that the working surface of the expanding remnant has a transverse size ζ m r + c s t co. The evolution of the burst changes when the second term dominates over the first. Each of these assumptions may be varied, but we believe the qualitative change in burst remnant evolution will remain over a wide range of possible beaming models. Removing assumption 4) is the only obvious way to turn off the dynamical effects of beaming, and even then observable breaks in the light curve are expected when Γ 1/ζ m. There are several models in the literature that use radiative rather than adiabatic models, dropping our second assumption. The case for radiative bursts depends on the efficiency with which relativistic shocks transfer bulk kinetic energy to magnetic fields and electrons, and I regard the validity of assumption 2) as an open question. For a closer examination of this issue, I refer the reader to papers by Vietri 1997a,b) and by Katz & Piran 1997a), who advocate radiative models; and to Waxman, Kulkarni, & Frail 1998), who defend the adiabatic model. Mészáros, Rees, & Wijers 1997) point out that the dynamical consequences Γ r 3 ) of radiative models depend on equipartition between protons, electrons, and magnetic fields being maintained at all times. Thus, a short electron cooling time will affect the afterglow radiation, but will not necessarily result in Γ r 3. Sari 1997) considers corrections to the adiabatic burst evolution for modest energy losses. Models that do not use assumption 1) have been discussed by Mészáros, Rees, & Wijers 1997) and Panaitescu, Mészáros, & Rees 1998). Finally, assumption 3) has been dropped by several authors Vietri 1997b; Mészáros, Rees, & Wijers 1997; Panaitescu, Mészáros, & Rees 1998) in favor of a more general power law density ρ r g. Such models complicate the beamed burst analysis and will change the form of Γr) relation but will leave intact the basic conclusion that Γr) changes qualitatively when Γ < 1/ζ m Dynamical Calculations: Numerical Integrations Given these assumptions, the full equations describing the burst remnant s evolution are f = 1 r r 2 Ω m r)ρr)dr, 2) M 0 0 Ω m πζ m + c s t co /ct) 2 πζ m + t co / 3t) 2, 3) Γ=Γ 0 +f)/ 1+2Γ 0 f+f 2 Γ 0 /2f, 4) t = r/c, t co = t 0 dt /Γ,

3 James E. Rhoads 3 and t =1+z) t 0 dt /2Γ 2. 5) Equation 4 is derived in Paczyński & Rhoads 1993) from conservation of energy and momentum, along with algebraic simplifications of equations 5 for the spherical case. The definition of t here includes the cosmological time dilation factor 1 + z) for a source at redshift z. Equation 3 is not strictly valid when ζ m > 1, but we will accept this deficiency since the error thereby introduced is not a dominant uncertainty in our results. These equations can be solved by numerical integration to yield fr), Γr), and t r). Figure 1 shows Γr) from such integrations for an illustrative pair of models one beamed, one isotropic) Dynamical Calculations: Analytic Integrations The most interesting dynamical change introduced by beaming is a transition from a power law Γ r 3/2 to an exponentially decaying regime Γ exp r/r Γ ). We will first give a derivation of the power law behavior Power Law Regime Consider the approximate evolution equations for the regime where a) 1/Γ 0 <f <Γ 0,sothatΓ Γ 0 /2f; and b) c s t co <ζ mrcorresponding to f < 9Γ 0 ζm). 2 df /dr π ζ m r) 2 2f ρ, dt co /dr M 0 c 2, Γ 0 dt /dr 1 + z) f. 6) cγ 0 The initial conditions are f =0andt=t co = t =0at r= 0. So we can easily integrate and obtain whence Γ f = πζ2 m 3 ρ M 0 r 3, 7) ) 1/2 ) 1/2 3M0 Γ 0 3E 0 2πζmr 2 3 = ρ 2πζmc 2 2 r 3 8) ρ ) 8πζ 2 1/2 t co = m ρ r 5/2 = 2 r 75E 0 5 cγ t =1+z) πζ2 m 12 9) cρ E 0 r 4 10) ) 1/8 ) 3/8 Γ=2 5/4 3E0 1+z πζm 2. 11) c2 ρ ct By making the substitutions πζ 2 m 4π and 1 + z) 1 in these results, we recover the evolution of a spherically symmetric burst remnant derived by Paczyński & Rhoads 1993) Exponential Regime To demonstrate the exponential behavior, consider the approximate evolution equations for the regime where a) 1/Γ 0 <f <Γ 0,sothatΓ Γ 0 /2f;andb)c s t co >ζ m r corresponding to f > 9Γ 0 ζ 2 m): df /dr π 2f c 2 s M t2 co ρ, dt co/dr 0 c 2, Γ 0 dt /dr 1 + z) f. 12) cγ 0 By forming the ratio df /dr)/dt co /dr) and isolating terms with f and with t co, it follows that π cc 2 fdf = 2 sρ Γ 0 t 2 M codt co π 0 3 c 3 ρ Γ 0 t 2 2 M codt co. 0 13) This is easily integrated to obtain f 3/2 = π Γ0 cc 2 sρ 8M0 ) t 3 co c 1 ), 14) where c 1 is a constant of integration. Using the initial conditions for the exponential regime derived below eqn ), one can show that the constant of integration is c 1 = 25E 0 ζm/4πρc 3 5 s), which becomes negligible once c s t co ζ m r. Equation 14 then becomes f t 2 co,andwe see from equations 12 and 4 that f, Γ,t co,andt will all behave exponentially with r in this regime. Retaining the constants of proportionality, we find f exp2r/r Γ ) where [ ) ] 2 1/3 1 c Γ 0 M 0 r Γ = = π ρ c s [ ] 1/3 E0 πc 2 s ρ. 15) Further algebra yields Γ exp r/r Γ ), t co f expr/r Γ ), and t f exp2r/r Γ ), so that Γ t 1/2. Thus, while the evolution of Γr) changes from a power law to an exponential at Γ 1/ζ m, the evolution of t r) changes similarly. The net result is that Γt )hasapower law form in both regimes, but with a break in the slope from Γ t 3/8 when Γ > 1/ζ m to Γ t 1/2 when Γ < 1/ζ m. The initial conditions for the exponential regime are approximately set by inserting the transition condition c s t co = ζ m ct into the evolution equations for the power law regime. Denoting the values at this break with the subscript b,wehavec s t co,b = ζ m ct b = ζ m r b, which we combine with equation 9 to obtain ) 1/3 75E0 r b = 8πρc 2 and f b = 25 s 8 c The corresponding values for Γ, t,andt co are c s ) 2 ζmγ ) Γ b = 2c s 1, 17) 5c ζ m ) 1/ /3 ) 1/3 c E0 t,b = 1+z) π 64 c s ρc 5 ζm 2 18), s ) 1/3 75E0 and t co,b = ζ m 8πρc 5 19) s

4 4 Submitted to The Astrophysical Journal, 1998; current revision 3/1999 The evolution in the exponential regime is then approximated by f = f b exp {2r r b )/r Γ } t = t,b exp {2r r b )/r Γ } Γ = Γ b exp { r r b )/r Γ } t co = t co,b exp {r r b )/r Γ } 20) A thought experiment that will help understand the onset of the exponential decay of Γ with radius is to consider the shape of a GRB remnant in a pressureless, uniform ambient medium at late times after all motions have become nonrelativistic). In the spherical case, the blast wave will leave behind a spherical cavity. In the beamed geometry, the cavity will be conical near the burster, but will change shape at the radius where the lateral expansion of the remnant becomes important. At this point, the cone flares, and the mass swept up per unit distance begins to grow faster than r 2. This corresponds to the onset of the exponential Γr) regime. The cone continues to become rapidly wider until it reaches the radius where the remnant becomes nonrelativistic. The final cavity resembles the bell of a trumpet. It is unclear whether such remnants would survive long enough to be observed in a realistic interstellar medium. 3. EMERGENT RADIATION The Lorentz factor Γ is not directly observable, and we ultimately want to predict observables like the frequency of peak emission ν,m, the flux density F ν,,m at ν,m,and the angular size θ of the afterglow. To do so, we need to introduce a model for the emission mechanism. We will restrict our attention to synchrotron emission, which is the leading candidate for GRB afterglow emission. We first consider the case of optically thin emission with a steep electron energy spectrum. This emission model is used in many recent afterglow models e.g. Waxman 1997a,b). We then repeat the calculation for the emission model of Paczyński & Rhoads 1993) General equations: Optically thin case Our dynamical model for burst remnant evolution gives the volume V and internal energy density u i of the ejecta as a function of expansion radius r. Detailed predictions of synchrotron emission require the magnetic field strength and the electron energy spectrum. We assume that the energy density in magnetic fields and in relativistic electrons are fixed fractions ξ B and ξ e of the total internal energy density. The magnetic field strength B follows immediately: B = 8πξ B u i. 21) N.b., we use the notation of Paczyński & Rhoads Some other authors have instead defined ξ B in terms of the magnetic field strength, such that B ξ B in their models; care must therefore be taken in comparing scaling laws under these alternative notations.) The electron energy spectrum requires additional assumptions. We first follow Waxman s 1997a,b) assumptions, to facilitate comparison of our results for beamed bursts with his for unbeamed bursts. In the frame of the expanding blast wave, the swept-up ambient medium appears as a relativistic wind having Lorentz factor Γ. We assume that the electrons from the ambient medium have their direction of motion randomized in the blast wave frame. Moreover, they may achieve some degree of equipartition with the protons. The typical random motion Lorentz factor γ e for the swept-up electrons in the blast wave frame is then in the range Γ < γ e < 0.5m p /m e )Γ. In terms of the energy density fraction in electrons, γ e ξ e m p /m e )Γ. We further assume that the electrons in the original ejecta mass are not heated appreciably, so that the number of relativistic electrons is N e = fm 0 /µ e m p )whereµ e is the mean molecular weight per electron) rather than 1 + f)m 0 /µ e m p ). We take the electron energy E to be distributed as a power law NE) =E p for E min < E < E max,wherene)deis the number of electrons with energies between E and E + de. Finally, we assume that p>2, so that the total electron energy E max E min ENE)dE is dominated by electrons with E E min,andγ e,peak γ e ξ e m p /m e )Γ E min /m e c 2 ). The optical depth to synchrotron self-absorption is assumed to be small at the characteristic synchrotron frequency corresponding to E min = γ e,peak m e c 2. In the comoving frame, this frequency is ν co,m = /4π) sin α γe,peak 2 eb/m ec) = /16)γe,peak 2 eb/m ec) Pacholczyk 1970, Rybicki & Lightman 1979), where the calculation of the mean pitch angle sin α = π/4 assumes an isotropic distribution of electron velocities and a tangled magnetic field. Wijers & Galama 1998) have integrated over the power law distribution of electron energies to show that the peak comoving frame frequency for a power law energy distribution becomes ν co,m E =3x p /4π) γe,peak 2 eb/m ec), where x p is a function of the power law index p, and where 0.64 >x p >0.45 for 2 <p<3. Below this peak frequency, the flux density rises as ν 1/3, while at higher frequencies it falls as ν α where α = p 1)/2 e.g., Rybicki & Lightman 1979, Pacholczyk 1970). Three additional breaks may occur, corresponding to the highest electron energy attained in the shock, the electron energy above which cooling is important, and the frequency where synchrotron self-absorption becomes important. We will comment on the cooling break below, and will ignore the other two breaks for the present. We first estimate the observer-frame frequency ν,m at which the spectrum peaks. This is ν,m 1+βcosθ) θ Γ ν co,m E /1 + z) 1 1+z 4Γ 3 3x p 4π γ2 e,peak eb m e c ) 2 1 x p m p eb ξ e 1+z π m e m e c Γ3 22) where the factor 1+βcosθ) Γ is the Lorentz transformation for frequency, and where β = 1 Γ 2 is the expansion velocity as a fraction of lightspeed c. θ is the angle between the velocity vector of radiating material and the photon emitted, as measured in the frame of the emitting matter. 1 + βcosθ) denotes an average weighted by the received intensity. We shall use the highly relativistic limit β 1 throughout this work, which leads to the result 1+βcosθ) =4/3 applied in the final lines of equation 22 Wijers & Galama 1998).

5 James E. Rhoads 5 We estimate the peak flux density following equations 19 and 25 of Wijers & Galama 1998). The basic equation is F ν,m, =ΓN e φ p 3e 3 B m e c 2 1+z Ω γ d 2. 23) Here φ p 3e 3 B/m e c 2 ) is the average comoving frame peak luminosity per unit frequency emitted by a single electron. The details of the average over pitch angle and electron energy are hidden in the factor φ p, which is a function of the electron energy distribution index p with range 0.59 <φ p <0.66 for 2 <p<3 Wijers & Galama 1998). The factor Γ accounts for the Lorentz transformation of flux density Wijers & Galama 1998). Distance and beaming effects enter through the factor 1/Ω γ d 2 ), where d is the luminosity distance to the burst, and Ω γ is the solid angle into which radiation is beamed. Finally, redshift affects the flux density by factor 1 + z) e.g. Weedman 1986). Our goal is to express F ν,m, purely in terms of the dynamical variables we calculated in section 2. The comoving frame synchrotron cooling time is t s = 6πm ec σ T γ e B 2 6πm e c σ T γ e,peak B 2, 24) where σ T = cm 2 is the Thompson crosssection e.g., Rybicki & Lightman 1979). To obtain the magnetic field strength B, we need the volume of the ejecta cloud, which will have transverse radius rζ m + c s t co and thickness c s t co, giving comoving volume V = πctζ m + c s t co ) 2 c s t co ). Under the approximation of negligible radiative losses, the internal energy is given by E i,co = E 0 /Γ. The comoving frame magnetic field strength is thus ) 1/2 8ξ B = B E 0 Γctζ m + c s t co ) 2. 25) c s t co ) The remaining pieces are trivial. Ω γ π ζ m +1/Γ) 2, and d and 1 + z are simply scale factors. We now have all the pieces of equations 22 and 23 expressed in terms of dynamical variables from section 2. This means that we can insert these formulae into our numerical integration code and calculate F ν,m, and ν,m as a function of t or of f, Γ,orr). In order to determine a light curve at fixed observed frequency, we combine the broken power law spectral shape described above with the calculated frequency and flux density of the spectral peak to determine the approximate flux density at the observed frequency and time. The cooling break and self absorption break cf. Sari, Piran, & Narayan 1998) are additional observed features in afterglow data. We do not treat either in detail here, but do we present a derivation of the cooling break behavior for beamed gamma ray bursts elsewhere Rhoads 1999b). These results are summarized below. We have not yet treated the self-absorption break. Self-absorption is important primarily at low frequencies, where scintillation can hamper light curve slope measurements. For a treatment of this regime in beamed bursts, see Sari, Piran, & Halpern 1999). Finally, we consider the evolution of the apparent angular size θ. In the spherical case or the power-law regime for a beamed burst, θ = r/γd θ ) t 5/8 where d θ is the angular diameter distance to the burst). In the exponential regime, θ is determined by the physical transverse size of the ejecta cloud rather than the beaming angle, but the difference is not dramatic because the physical size increases at c s c. The result is therefore θ c s t co /d θ 1/Γ t 1/2. There is also an intermediate regime, valid for the brief time when 1/Γ >ζ m >c st co )/ct). In this case, θ ζ m r t 1/4. If the exponential regime did not happen at all, this behavior would continue for all Γ < 1/ζ m Analytic Results: Optically thin case In the limiting cases where one of the terms in the transverse size ζ m ct+c s t co is dominant and the other negligible, we can derive analytic expressions for F ν,m, and ν,m as functions of observed time t and the physical parameters of the fireball. We begin with the early time case, and show that its light curve is observationally indistinguishable from that of an isotropic burst Power Law Regime We first determine the comoving magnetic field in this regime by inserting ζ m ct c s t co into equation 25 to obtain B = 21/4 5π c 7/8 3 3/8 c 1/2 s ξ 1/2 E0 B πζm 2 ) 1/8 ) 3/8 ρ1 + z). 26) Inserting this result into equation 22, we find ν,m = 21/4 5 1/2 ) 2 3 3/8 π x pξeξ 2 1/2 mp e 1/2 B m e m e c 1/8 c 1/2 s ) 1/8 ) 3/8 E0 ρ Γ z) 5/8 27) πζ 2 m = 51/2 2 x pξeξ 2 1/2 7/2 B E0 πζ 2 m t mp m e t ) 2 e m e c 2 c 1/2 s ) 1/2 t 3/2 1 + z) 1/2 28) where we have used equation 11 to eliminate Γ in the last line. For the cooling break, we obtain ν,cool t 1/2 Rhoads 1999b; Sari et al 1999). Turning our attention to F ν,m,,wefirstneedthenumber N e of radiating electrons in terms of t.forthepower law regime, this becomes N e = 23/2 πζ 2 m ρ ) 1/4 ) 3/4 E0 t. 29) 3 1/4 µ e m p c1 + z) Combining this with equations 23, 26, and the appropriate limiting form of Ω γ, we find F ν,m, = 10π φ pξ 1/2 e 3 B c ρ 1/2 E 0 1+z µ e m p m e c 3 c s πζm 2 d 2. 30) Note that this result is independent of t.

6 6 Submitted to The Astrophysical Journal, 1998; current revision 3/1999 Apart from small differences in the numerical coefficients, our results for the power law regime are essentially the same as the results that Waxman 1997a,b) and Wijers and Galama 1998) obtained for isotropic bursts. Differences between our results and Waxman s are primarily because we have adopted the more precise treatment of the synchrotron peak frequency presented by Wijers and Galama 1998), while differences between our results and those of Wijers and Galama stem from a slightly different way of calculating the comoving frame magnetic field Exponential Regime When c s t co ζ m ct, we are in the regime where Γ, t, etc. all behave exponentially with radius section 2.2.2). We first rewrite the scalings from equation 20 as t /t,b = exp{2r r b )/r Γ } f/f b = exp{2r r b )/r Γ } = t /t,b Γ/Γ b = exp{ r r b )/r Γ } = t /t,b ) 1/2 t co /t co,b = exp{r r b )/r Γ } = t /t,b ) +1/2. 31) We next determine the comoving frame magnetic field in the appropriate limit: ) 1/2 ) 1/2 8ξB E 0 B = Γ 1/2 b t 3/2 t co,b. 32) t,b c 3 s Combining this result with equation 22 allows us to determine the peak frequency as 1 2 3/2 ) 2 ν,m = 1+z π x pξeξ 2 1/2 mp e B m e m e c ) 2 E 1/2 c s t co,b ) 3/2 t. 33) 0 Γ 5/2 b t,b Substituting for the exponential regime initial conditions from equations then yields ν,m = 1 1+z e m e cs c 2 11/2 3 1/2 5 7/2 π x pξ 2 1/2 e ξ1/2 B ) 7/2 ρ 1/2 ζ 4 m t = 1+z) 31/6 5 11/6 2 13/2 π x pξ 2 7/6 e ξ 1/2 B e m e cs c ) 3/2 E 2/3 0 ρ 1/6 c 10/3 s ) 2 mp m e ) 2 34) t,b ) 2 mp m e t 2. 35) The observed cooling break frequency ceases to evolve in this regime: ν,cool t 0 Rhoads 1999b; Sari et al 1999). Turning now to the amplitude of the spectral peak, we combine N e = 25 E 0 ζm 2 t 8 c 2 s µ 36) em p t,b and Ω γ = πγ 2 = πγ 2 b t /t,b ) 37) with equations 23 and 32 to obtain F ν,m, = 31/ /2 π φ pξ 1/2 e 3 E 3/2 B m e m p µ e c 2 c 2 s c s t co,b ) 3/2 Γ 5/2 b t t,b 0 ζ 2 m 38) ) 1 1+z d 2.39) Substituting the initial conditions for the exponential regime, this becomes 32π F ν,m, = 125 φ pξ 1/2 e 3 cs ) 3/2 B c 3 40) m p m e µ e c E ) 0 ρ 1 t 1+z πζ 2 m t,b = 31/3 5 7/6 2 7/2 π 5/6φ pξ 1/2 e 3 cs ) 1/2 B 41) c 3 m p m e µ e c E4/3 0 ρ 1/6 1 + z) 2 c 5/3 d s 2 t 1. At t = t,b, equations 34 and 40 differ from equations 28 and 30 by factors of order unity. This difference is not worrying since our analytic approximations are not expected to be particularly accurate in the transition between the two limiting cases. Numerical correction factors to the coefficients of equations 34 and 40 can be derived from numerical integrations. Such factors are presented in section below TV Dinner Equations We now pause a moment to consolidate our results so far and express the key equations in terms of fiducial parameter values 2. We begin with equations 28 and 30. These become ) 1/2 ν,m = z) 1/2 cs c/ 3 xp ) ) 2 ) 1/2 ξ e ξb 42) and E0 /10 53 erg ζ 2 m/4 ) 1/2 t day d ) 3/2 Hz ) cs ) 1/2 φp F ν,m, =111 + z) 0.63 c/ 43) 3 ) 1/2 ) ξb 1.3 E0 /10 53 ) erg 0.1 ζm/4 2 µ e ρ g/ cm 3 ) 1/2 d 4.82 Gpc ) 2 mjy. The observed time corresponding to the transition between the power law and exponential regimes is ) 8/3 cs t,b = z) c/ E0 /10 53 ) 1/3 erg 3 ζm/4 2 ) 1/3 ) 8/3 ρ ζm g/ cm 3 days. 44) We call these TV dinner equations because numerical values for physical constants have been inserted, so they are ready to use without further preparation.

7 James E. Rhoads 7 Thereafter, the frequency and flux density at the spectral peak are characterized by equations 28 and 40. Numerical integrations show that modest correction factors ɛ ν 0.74 and ɛ F 0.7 should be applied to these two equations at late times to compensate for approximations in the initial conditions see section below). These have been incorporated in the following three equations. The observed frequency of the spectral peak at the time of the break is ν,m,b = 1+z ) 2 ξe ξb ɛν x p ) c s ) 1 2 ρ g/ cm 3 c/ 3 ) 1 ) 7/2 45) 2 ζ m 0.1 ) 4 Hz. The subsequent evolution is given by ) 2 t ν,m = ν,m,b 46) t,b and ) ) 3/2 ) 1/2 ɛf φ p cs F ν,m, = c/ ξb 47) ) 1.3 E0 /ζ 2 ) ) 1/2 m ρ µ e erg/ g/ cm 3 ) 2 ) 1 d t 1 + z) mjy Gpc t,b Finally, for completeness, we include our result for ν,cool from Rhoads 1999b: [ ν,cool = t /t,b ) 1/ ] Hz ) ) 17/6 ) 3/2 1 cs 1+z c/ ξb 48) ) ρ cm 3 5/6 E0 /10 53 ) 2/3 ) 4/3 erg ζm g ζm 2 / Note that this equation already interpolates over the break time t,b ; the interpolation was derived in the fashion suggested in section below Putting the Pieces Together An accurate description of the behavior in the transition between the power law and exponential regimes can be obtained numerically. We first note that there is a single characteristic observed time t,b given by equation 18 and flux level F ν,m,,b F ν,m, t t,b ) given by equation 30. If we use these as our basic time and flux units, and denote the observed time and peak flux scaled to these units as t and F ν,m,, there is a unique Fν,m, t )relation. This is plotted in figure 2. Similarly, we can define the characteristic frequency ν,m,b in the problem to be given by equation 28 evaluated at t,b, and ν,m to be the frequency scaled by this value. Then we can again obtain a unique relation ν,m t ), which is shown in figure 3. At late times, the numerical integrations yield a flux density that is a factor ɛ F 0.7 smallerthaninequation 40, and a frequency of peak emission that is a factor ɛ ν 0.74 smaller than in equation 34. This is presumably due to the approximate initial conditions used for the exponential regime evolution. These initial conditions are obtained by applying an asymptotic approximation outside its range of validity, and it should not be surprising if this procedure introduces some error. We suggest below that this error may be corrected empirically. To obtain predictions for a given set of model parameters from these dimensionless curves, we need only 1) determine numerically the values of t,b, F ν,m,,b, and ν,m,b ; and 2) determine the time interval over which our assumption 1/Γ 0 <f <Γ 0 remains valid. The early behavior, before the ejecta accrete a dynamically important amount of ambient medium i.e., f<1/γ 0 ) is unlikely to be observed at long wavelengths, since it is over within a fraction of a second for reasonable burst parameters. We therefore consider only the end condition, f Γ 0. This happens at t = t,f,where t,f Γ 0 t,b /f b 8 cs ) c ζm 2 t,b. 49) At later times, our assumptions that Γ > 2andβ 1 break down, and the behavior of the fireball changes again. Such changes may be relevant to the radio behavior of gamma ray burst afterglows, but we will not consider them here Empirical Interpolations To obtain a readily calculated burst behavior around time t,b, we can interpolate between the asymptotic behaviors for earlier and later times. We do this first for F ν,m, and then for ν,m. We use interpolants of the form g =g1 κ +ɛg2 κ ) 1/κ,whereg 1 and ɛg 2 represent limiting behaviors of an arbitrary function g for early and late times. The exponent κ determines the smoothness of the transition between the limiting behaviors. The scalar ɛ is introduced so that the numerically derived correction factors to the late-time asymptotic results can be applied. For F ν,m,, the asymptotic behaviors are F ν,m, constant and F ν,m, t. We work with the scaled quan- 1 tities defined in section 3.2.4, so that the break between the two asymptotic behaviors is expected for log t ) 0. We set g 1 = F ν,m,,b. We use equation 40 for g 2,andset the correction factor ɛ =0.7. Finally, we choose κ =0.4. The resulting interpolation is plotted atop the numerical integration results in figure 2. The asymptotic behaviors of ν,m are t 3/2 and t 2. In this case, we have taken g 1 from equation 28. For g 2, we take equation 34, and set ɛ =0.74. Here we find κ = 5/6 works well. This interpolation is shown in figure Light Curves: Optically thin case The afterglow light curve at fixed observed frequency is obtained by combining the predicted behavior of ν,m and F ν,m, with the spectrum for a truncated power law electron energy distribution. We use the analytic results of section 3.2. Then we find four generic behaviors, depending on whether the frequency is above or below ν,m

8 8 Submitted to The Astrophysical Journal, 1998; current revision 3/1999 and whether the time is earlier or later than t,b. These are t 1/2 t <t,b ; ν <ν,m t ) t 3p 1)/4 F ν, t <t,b ; ν >ν,m t ) t 1/3 t >t,b ; ν <ν,m t ) t p t >t,b ; ν >ν,m t ) 50) Here p is the electron energy spectrum slope and α = p 1)/2 is the high frequency spectral slope, as usual. Note particularly how steep the light curve becomes for t >t,b and ν >ν,m t ). Three representative light curves are shown in figure 4. These have been derived by combining the empirically interpolated ν,m and F ν,m, curves with the broken power law spectrum. Note that the rollover at the beaming transition log t ) 0) is rather slow, so that observed behavior will be intermediate between the asymptotic power laws of equations 50 for a considerable time. This slow rollover is in part due to the compound nature of the break. The light curve decay begins to accelerate as soon as we can see the edge of the jet, when Γ < 1/ζ m.the additional steepening when dynamical effects of beaming become important occurs slightly later, when Γ < Γ b 0.23/ζ m cf. equation 17) cf. Panaitescu & Mészáros1999 for additional discussion of this point). Equation 50 assumes ν,abs <ν <ν,cool throughout. Here ν,abs is the self-absorption frequency, measured in the observer s frame.) If we now include the cooling break, we obtain the additional light curve behaviors derived in Rhoads 1999b) { 1/2 3p/4 t F ν, t <t,b ; ν >ν,cool t ) t p t >t,b ; ν >ν,cool t ) 51) wherewehavealsoassumedthatν,m <ν,cool. These behaviors are not shown in figure 4, but were used to fit the light curve of GRB with beamed afterglow models Rhoads 1999b) Light Curves: Optically thin case without sideways expansion It remains possible to constrain gamma ray burst beaming by looking for light curve breaks even in the case where lateral expansion of the evolving burst remnant is unimportant. This corresponds to dropping our fourth assumption from section 2. In this case, the dynamical evolution follows power law behavior section 2.2.1) throughout, but the emergent radiation is diluted relative to the spherical case by a factor Γ 2 ζm 2 once Γ < 1/ζ m. For an adiabatic evolution, we have Γ 2 t 3/4. The power law exponents for the light curve in this regime become 1/4 forν,abs <ν<ν,m, 3p/4 for ν,m <ν<ν,cool,and 1/4 3p/4 forν,cool <ν. The most plausible mechanism for quenching lateral expansion is a strongly radiative remnant, so it is more relevant to examine this regime with radiative dynamics. Then Γ 2 t 6/7, and all the light curve exponents decrease by an additional 3/28, becoming 5/14 for ν,abs <ν<ν,m, 3/28 3p/4 forν,m <ν<ν,cool, and 5/14 3p/4 forν,cool <ν. While these slope changes are less dramatic than those in section 3.2.6, they would be strong enough to detect in afterglow light curves with reasonably large time coverage and good photometric accuracy Optically Thick Case We now consider briefly the electron energy distribution model of Paczyński & Rhoads 1993). This model differs from that of the preceding sections in a few ways. First, the electron power law index was fixed at p =2 to avoid strong divergences in the total energy density in electrons. Second, the minimum electron energy E min was taken to be sufficiently small that emission from electrons with E = E min was always in the optically thick regime. Under these circumstances, there is a single break in the electron energy spectrum at the frequency corresponding to optical depth τ = 0.35 cf. Pacholczyk 1970), with spectral slope ν 5/2 below the break and ν 1/2 above. The magnetic field behavior is the same in this model and more recent ones. Combining this electron behavior with the power law regime dynamical model reproduces the scalings from Paczyński & Rhoads 1993), namely and ν,m E 1/3 0 ρ 1/3 t 2/3 52) F ν,,m E 7/8 0 ρ 1/8 ν 5/8,m E13/12 0 ρ 1/3 t 5/12. 53) If we use instead the exponential regime dynamical model, we find ν,m t 1 and F ν,,m t 3/2. 54) Readers interested in the precise numerical coefficients for these relations are referred to Paczyński & Rhoads 1993) for the spherical case. For the beamed case, numerical results may be found by applying the Paczyński & Rhoads 1993) results at the transition between the power law and exponential regimes, and continuing the evolution using equations 54. The light curve for this electron model then becomes t 5/4 t <t,b ; ν <ν,m t ) F ν, t 3/4 t <t,b ; ν >ν,m t ) t 1 55) t >t,b ; ν <ν,m t ) t 2 t >t,b ; ν >ν,m t ) The behavior here at frequencies ν >ν,m t )isthe same as in equations 50 with p = 2. However, ν,m has a different meaning in the two models. From these results, we see that the substantial changes in the observable behavior of a beamed burst are not dependent on the precise nature of the electron energy distribution. 4. DISCUSSION We now put mathematics aside to recapitulate our results and to discuss their implications for the interpretation of afterglow observations. We have shown that the dynamics of a gamma ray burst remnant change qualitatively when the remnant s Lorentz

9 James E. Rhoads 9 factor Γ drops below the reciprocal opening angle 1/ζ m of the ejecta. Before this time, the Lorentz factor behaves as a power law in radius. Afterwards, the Lorentz factor decays exponentially with radius. The change occurs because lateral expansion of the ejecta cloud increases the rate at which additional material is accreted. Such lateral expansion is prohibited by symmetry in the spherical case. When the remnant enters this exponential regime, the relation between the observed spectrum and the observed light curve changes. Inferences about the electron energy spectrum in afterglows come from the light curve decay rate and spectral slope. The general agreement between the two methods has been taken as a confirmation of the spherically symmetric) fireball model Wijers, Rees, & Mészáros 1997; Waxman 1997a). The light curve decline at frequencies above the spectral peak becomes very steep t p,wherepis the index of the electron energy spectrum) once the burst dynamics enter the exponential regime. Reconciling this relation with the observed decays 1 >dlogf ν, )/d logt ) > 1.5) would require an extremely flat electron energy spectrum, and consequently a very blue spectral energy distribution. This was not seen in early observed spectral energy distributions see Wijers et al 1997 for GRB ; Reichart 1997 and Sokolov et al 1997 for GRB ; and Reichart 1998 for GRB ). We infer that GRBs , , and were probably not in the exponential regime during their observed optical afterglows. GRB provides a possible, though dubious, counterexample. There is one image suggesting a counterpart magnitude R 19.5) on December Castro-Tirado et al 1997). Later images show no corresponding source, requiring a decay at least as fast as t 2.5 Djorgovski et al 1998). This is consistent with a t p decay for typical values of p. However, this explanation remains speculative, since there is no second image confirming the proposed counterpart. Subsequent afterglows have provided a more hopeful picture for practical application of beaming models. In particular, GRB shows a break that is quite possibly due to beaming e.g., Castro-Tirado et al 1999; Kulkarni et al 1999), and comparison of the spectral slope and decay slope for GRB gives better agreement for beamed than for spherical regime models Sari et al 1999). In the case of GRB , we can place a stringent limit on the beaming angle in the context of our model. The optical light curve extends to 100 days after the burst and does not depart drastically from a single power law after day 2 Pedersen et al 1998); thus, no transition to the exponential regime occurred during this time. As already noted, the spectral slope and light curve law decay rate are in fair agreement for the spherical case, and poor agreement for the beamed case. The radio light curve furnishes the last critical ingredient. Goodman 1997) pointed out that diffractive scintillation by the Galactic interstellar medium is expected in early time radio data, and that this scintillation will stop when the afterglow passes a critical angular size. By comparing this characteristic size with the time required for the scintillations to die out, one can measure the burst s expansion rate. This test has been applied Frail et al 1997; Waxman, Kulkarni, & Frail 1998) and shows that Γ < 2att 14 days. Thus, no power-law break to faster decline is observed at Γ > 2, and we infer that GRB was effectively unbeamed ζ m > 1/2). This rough derivation is borne out by detailed fitting of beamed afterglow models to the GRB light curve, which yields the same beaming limit ζ m > 1/2 radian Rhoads 1999b). This conclusion, combined with the GRB redshift limit z Metzger et al 1997), immediately implies a minimum energy for the burst. This burst was detected as BATSE trigger 6225, and the total BATSE fluence was 3.1±0.2)10 6 erg/ cm 2 over the range kev Kouveliotou et al 1997). The gamma ray emission alone therefore implies E 0 > Ω/4π)erg > erg. Here we have based the luminosity distance on an Ω = 0.2, Λ = 0, H 0 =70km/s/Mpc cosmology, and applied the beaming angle limit ζ m > 0.5 radian in the second inequality. This conclusion is of course modeldependent and might change if our assumptions about the blast wave physics or beaming geometry are badly wrong. We will discuss possible ways to reduce the energy requirements of GRB while retaining consistency with the afterglow data in section 4.1 below. If the beaming angle ζ m is substantially variable from burst to burst, it is possible that some bursts enter the rapid decay phase before the spectral peak passes through optical wavelengths. Present data suggests that this is indeed the case; GRB is best fit by assuming exponential regime behavior Sari et al 1999), while GRB appears to be a transition case with a break observed in the optical light curve e.g. Castro-Tirado et al 1999, Kulkarni et al 1999, Sari et al 1999). The resulting rapid decay could then explain some of the optical nondetections of well studied GRBs such as Groot et al 1997). Alternatively, for characteristic beaming angles 1 ζ m > 0.1, we would expect beaming to become dynamically important between the time of peak optical and radio afterglow. This would then help explain the paucity of radio afterglows, which unlike optical afterglows cannot be hidden by dust in the burster s environment. There is some evidence that the radio emission involves a different process, or at least a different electron population, from the optical and X-ray afterglows: The peak flux density in GRB did not follow a single power law with wavelength as it ought to under the simplest fireball models Katz & Piran 1997b). The transition in light curve behavior at Γ 1/ζ m is also important for blind afterglow searches. Such searches would look for afterglows not associated with detected gamma ray emission. A much higher event rate for afterglows than for bursts is a natural consequence of beamed fireball models, since the afterglow emission peaks at lower bulk Lorentz factors than the gamma ray emission does. Comparison of event rates at different wavelengths can therefore constrain the ratio of beaming angles at those wavelengths Rhoads 1997a). However, we will only see the afterglow if either a) we are within angle ζ m of the burst s symmetry axis, and therefore could also see the gamma ray burst, or b) the Lorentz factor has decayed to Γ < 1/ζ m and the afterglow light curve has entered its steep decay phase. We have already argued that GRB , GRB , and GRB were not in this steep decay phase based on the comparison of light curves and spectral slopes. It follows that if blind

10 10 Submitted to The Astrophysical Journal, 1998; current revision 3/1999 afterglow searches find a population of afterglows not associated with observed gamma rays, those afterglows will exhibit a steeper light curve decay than did the 1997 afterglows. The efficiency for detecting such rapidly fading orphan afterglows will be substantially lower than the efficiency estimated from direct comparison with sphericalregime afterglows. Other models of beamed gamma ray bursts are possible. In particular, we have assumed a hard-edged jet, where the mass and energy emitted per unit solid angle are constant at small angles and drop to zero as a step function at large angles. Profiles in which these quantities decrease smoothly to zero may be more realistic. Whether these differ importantly from the model presented here depends on whether most of the energy is emitted into a central core whose properties vary slowly across the core. Layered jet models in which most of the kinetic energy from the fireball is carried by material with a low Lorentz factor can have substantially different afterglow light curves from either the spherically symmetric case or the hard-edged jet case. This is because the afterglow emission can be dominated by outer layers where the initial Lorentz factor is high enough to yield optical emission during ejecta deceleration, but insufficient to yield gamma rays. The afterglow is thereby effectively decoupled from the gamma ray emission, and it becomes harder to predict one from the other. Such models have been explored by several groups e.g., Mészáros& Rees 1997b; Mészáros, Rees, & Wijers 1997; Paczyński 1997). A similar decoupling of the gamma-ray and afterglow properties can be produced in the spherical case by allowing inner shells of lower Lorentz factor and larger total mass and energy to follow the initial high-γ ejecta Rees & Mészáros 1998). It is possible to approximate the afterglow from a layered jet by a superposition of hard-edged jets. For this to be reasonably accurate, the outer layers should have Lorentz factors substantially below those of the inner layers, and opening angles and energies substantially above those of the inner layers Energy Requirements for GRB We now consider how our model will change if we vary some of the basic assumptions. Our primary concern is to determine whether the minimum energy required to power the GRB afterglow can be reduced substantially below the requirements derived from a spherical adiabatic fireball model expanding into a homogeneous medium. We will therefore sometimes err on the side of extreme model assumptions chosen to minimize the energy needs. In order to declare a model consistent with the data, we require that either 1) there be no break in the light curve or spectrum around Γ 1/ζ m, or 2) the break occurs early before t 2 days) and the late time light curve shows a slow decline even for spectral slopes as red as those observed. The first requirement is physically implausible. Even in the absence of the dynamical effects reported above, so long as the afterglow is from relativistically moving and decelerating material, its flux will scale with an extra factor Γ 2 once Γ > 1/ζ m. Since Γ decreases with time, a break is generally expected, though perhaps it could be avoided with sufficient fine-tuning of the model. The second possibility is more interesting. It requires us to construct a model where factors besides beaming contribute relatively little to the decay of F ν,m, with t, or where the observed spectrum does not directly tell us about the electron energy distribution. A burst expanding into a cavity such that ρ increases with r) might give a slow decay, while a sufficiently large dust column density would give a red spectrum despite a flat electron energy distribution cf. Reichart 1997, 1998). However, both would require some degree of fine tuning. The dustreddened spectra would deviate measurably from pure power-laws given good enough data, but the present data are probably equally consistent with both pure and reddened power law spectra. Certainly such reddened beaming models would imply little correlation between observed spectral slopes and light curve decays, since the dust column density could vary wildly from burst to burst. This hypothesis is somewhat ad hoc, but is consistent with present data and is supported by other circumstantial evidence linking GRBs to dust and star forming regions e.g. Groot et al 1997; Paczyński 1998). At present, then, it appears the most viable way of reconciling beamed fireball scenarios with the 1997 afterglow data. We now discuss a few variations of fireball models in greater detail Radiative case We first consider the behavior of a radiative regime fireball. In this regime, the internal energy of the fireball is low, since it is converted to photons and radiated away. The largest implications for beaming are when the internal energy density is so low that c s c. In this case, the lateral expansion that leads to the exponential regime of burst remnant evolution in the adiabatic case is unimportantly small. We assume this low sound speed through much of the following discussion. We assume that energy in magnetic fields and protons is transferred to electrons in the burst remnant on a remnant crossing time t co ). The electrons are assumed to maintain a power law energy spectrum, with a large E max whose precise value is determined by the requirement that the burst radiate its internal energy efficiently. Under these circumstances, the Lorentz factor scales as Γ r 3 and the comoving frame internal energy E int of the remnant follows the evolution de int Γρc 2 πζm 2 dr r2 E int dt co t co dr πρc2 ζm 2 Γr3 ) 4E int. r r 56) This admits a solution of the form E int πζm 2 ρc2 Γr 3 ) c 2 r 4 )/4 wherec 2 is a constant of integration. At late times, we throw away the c 2 r 4 term, which becomes negligible. The result then becomes E int Γr 3 )πζ 2 m ρc2 /4 57) which is constant since Γ r 3 in this regime. If the sound speed becomes negligibly small at some point in the burst remnant evolution, then the volume of the shell scales as V r 2 thereafter. The magnetic field then scales as B r 1, based on constant E int. The observed peak frequency scales as ν,m Γ 3 B r 10 t 10/7.